libm/e_jn.c

   1 /* @(#)e_jn.c 5.1 93/09/24 */
   2 /*
   3  * ====================================================
   4  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
   5  *
   6  * Developed at SunPro, a Sun Microsystems, Inc. business.
   7  * Permission to use, copy, modify, and distribute this
   8  * software is freely granted, provided that this notice
   9  * is preserved.
  10  * ====================================================
  11  */
  12
  13 #if defined(LIBM_SCCS) && !defined(lint)
  14 static char rcsid[] = "$NetBSD: e_jn.c,v 1.9 1995/05/10 20:45:34 jtc Exp $";
  15 #endif
  16
  17 /*
  18  * __ieee754_jn(n, x), __ieee754_yn(n, x)
  19  * floating point Bessel's function of the 1st and 2nd kind
  20  * of order n
  21  *
  22  * Special cases:
  23  *      y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal;
  24  *      y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal.
  25  * Note 2. About jn(n,x), yn(n,x)
  26  *      For n=0, j0(x) is called,
  27  *      for n=1, j1(x) is called,
  28  *      for n<x, forward recursion us used starting
  29  *      from values of j0(x) and j1(x).
  30  *      for n>x, a continued fraction approximation to
  31  *      j(n,x)/j(n-1,x) is evaluated and then backward
  32  *      recursion is used starting from a supposed value
  33  *      for j(n,x). The resulting value of j(0,x) is
  34  *      compared with the actual value to correct the
  35  *      supposed value of j(n,x).
  36  *
  37  *      yn(n,x) is similar in all respects, except
  38  *      that forward recursion is used for all
  39  *      values of n>1.
  40  *
  41  */
  42
  43 #include "math.h"
  44 #include "math_private.h"
  45
  46
  47 #ifdef __STDC__
  48 static const double
  49 #else
  50 static double
  51 #endif
  52 invsqrtpi=  5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */
  53 two   =  2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */
  54 one   =  1.00000000000000000000e+00; /* 0x3FF00000, 0x00000000 */
  55
  56 #ifdef __STDC__
  57 static const double zero  =  0.00000000000000000000e+00;
  58 #else
  59 static double zero  =  0.00000000000000000000e+00;
  60 #endif
  61
  62 #ifdef __STDC__
  63         double attribute_hidden __ieee754_jn(int n, double x)
  64 #else
  65         double attribute_hidden __ieee754_jn(n,x)
  66         int n; double x;
  67 #endif
  68 {
  69         int32_t i,hx,ix,lx, sgn;
  70         double a, b, temp=0, di;
  71         double z, w;
  72
  73     /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
  74      * Thus, J(-n,x) = J(n,-x)
  75      */
  76         EXTRACT_WORDS(hx,lx,x);
  77         ix = 0x7fffffff&hx;
  78     /* if J(n,NaN) is NaN */
  79         if((ix|((u_int32_t)(lx|-lx))>>31)>0x7ff00000) return x+x;
  80         if(n<0){
  81                 n = -n;
  82                 x = -x;
  83                 hx ^= 0x80000000;
  84         }
  85         if(n==0) return(__ieee754_j0(x));
  86         if(n==1) return(__ieee754_j1(x));
  87         sgn = (n&1)&(hx>>31);   /* even n -- 0, odd n -- sign(x) */
  88         x = fabs(x);
  89         if((ix|lx)==0||ix>=0x7ff00000)  /* if x is 0 or inf */
  90             b = zero;
  91         else if((double)n<=x) {
  92                 /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
  93             if(ix>=0x52D00000) { /* x > 2**302 */
  94     /* (x >> n**2)
  95      *      Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
  96      *      Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
  97      *      Let s=sin(x), c=cos(x),
  98      *          xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
  99      *
 100      *             n    sin(xn)*sqt2    cos(xn)*sqt2
 101      *          ----------------------------------
 102      *             0     s-c             c+s
 103      *             1    -s-c            -c+s
 104      *             2    -s+c            -c-s
 105      *             3     s+c             c-s
 106      */
 107                 switch(n&3) {
 108                     case 0: temp =  cos(x)+sin(x); break;
 109                     case 1: temp = -cos(x)+sin(x); break;
 110                     case 2: temp = -cos(x)-sin(x); break;
 111                     case 3: temp =  cos(x)-sin(x); break;
 112                 }
 113                 b = invsqrtpi*temp/sqrt(x);
 114             } else {
 115                 a = __ieee754_j0(x);
 116                 b = __ieee754_j1(x);
 117                 for(i=1;i<n;i++){
 118                     temp = b;
 119                     b = b*((double)(i+i)/x) - a; /* avoid underflow */
 120                     a = temp;
 121                 }
 122             }
 123         } else {
 124             if(ix<0x3e100000) { /* x < 2**-29 */
 125     /* x is tiny, return the first Taylor expansion of J(n,x)
 126      * J(n,x) = 1/n!*(x/2)^n  - ...
 127      */
 128                 if(n>33)        /* underflow */
 129                     b = zero;
 130                 else {
 131                     temp = x*0.5; b = temp;
 132                     for (a=one,i=2;i<=n;i++) {
 133                         a *= (double)i;         /* a = n! */
 134                         b *= temp;              /* b = (x/2)^n */
 135                     }
 136                     b = b/a;
 137                 }
 138             } else {
 139                 /* use backward recurrence */
 140                 /*                      x      x^2      x^2
 141                  *  J(n,x)/J(n-1,x) =  ----   ------   ------   .....
 142                  *                      2n  - 2(n+1) - 2(n+2)
 143                  *
 144                  *                      1      1        1
 145                  *  (for large x)   =  ----  ------   ------   .....
 146                  *                      2n   2(n+1)   2(n+2)
 147                  *                      -- - ------ - ------ -
 148                  *                       x     x         x
 149                  *
 150                  * Let w = 2n/x and h=2/x, then the above quotient
 151                  * is equal to the continued fraction:
 152                  *                  1
 153                  *      = -----------------------
 154                  *                     1
 155                  *         w - -----------------
 156                  *                        1
 157                  *              w+h - ---------
 158                  *                     w+2h - ...
 159                  *
 160                  * To determine how many terms needed, let
 161                  * Q(0) = w, Q(1) = w(w+h) - 1,
 162                  * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
 163                  * When Q(k) > 1e4      good for single
 164                  * When Q(k) > 1e9      good for double
 165                  * When Q(k) > 1e17     good for quadruple
 166                  */
 167             /* determine k */
 168                 double t,v;
 169                 double q0,q1,h,tmp; int32_t k,m;
 170                 w  = (n+n)/(double)x; h = 2.0/(double)x;
 171                 q0 = w;  z = w+h; q1 = w*z - 1.0; k=1;
 172                 while(q1<1.0e9) {
 173                         k += 1; z += h;
 174                         tmp = z*q1 - q0;
 175                         q0 = q1;
 176                         q1 = tmp;
 177                 }
 178                 m = n+n;
 179                 for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t);
 180                 a = t;
 181                 b = one;
 182                 /*  estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
 183                  *  Hence, if n*(log(2n/x)) > ...
 184                  *  single 8.8722839355e+01
 185                  *  double 7.09782712893383973096e+02
 186                  *  long double 1.1356523406294143949491931077970765006170e+04
 187                  *  then recurrent value may overflow and the result is
 188                  *  likely underflow to zero
 189                  */
 190                 tmp = n;
 191                 v = two/x;
 192                 tmp = tmp*__ieee754_log(fabs(v*tmp));
 193                 if(tmp<7.09782712893383973096e+02) {
 194                     for(i=n-1,di=(double)(i+i);i>0;i--){
 195                         temp = b;
 196                         b *= di;
 197                         b  = b/x - a;
 198                         a = temp;
 199                         di -= two;
 200                     }
 201                 } else {
 202                     for(i=n-1,di=(double)(i+i);i>0;i--){
 203                         temp = b;
 204                         b *= di;
 205                         b  = b/x - a;
 206                         a = temp;
 207                         di -= two;
 208                     /* scale b to avoid spurious overflow */
 209                         if(b>1e100) {
 210                             a /= b;
 211                             t /= b;
 212                             b  = one;
 213                         }
 214                     }
 215                 }
 216                 b = (t*__ieee754_j0(x)/b);
 217             }
 218         }
 219         if(sgn==1) return -b; else return b;
 220 }
 221
 222 #ifdef __STDC__
 223         double attribute_hidden __ieee754_yn(int n, double x)
 224 #else
 225         double attribute_hidden __ieee754_yn(n,x)
 226         int n; double x;
 227 #endif
 228 {
 229         int32_t i,hx,ix,lx;
 230         int32_t sign;
 231         double a, b, temp=0;
 232
 233         EXTRACT_WORDS(hx,lx,x);
 234         ix = 0x7fffffff&hx;
 235     /* if Y(n,NaN) is NaN */
 236         if((ix|((u_int32_t)(lx|-lx))>>31)>0x7ff00000) return x+x;
 237         if((ix|lx)==0) return -one/zero;
 238         if(hx<0) return zero/zero;
 239         sign = 1;
 240         if(n<0){
 241                 n = -n;
 242                 sign = 1 - ((n&1)<<1);
 243         }
 244         if(n==0) return(__ieee754_y0(x));
 245         if(n==1) return(sign*__ieee754_y1(x));
 246         if(ix==0x7ff00000) return zero;
 247         if(ix>=0x52D00000) { /* x > 2**302 */
 248     /* (x >> n**2)
 249      *      Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
 250      *      Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
 251      *      Let s=sin(x), c=cos(x),
 252      *          xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
 253      *
 254      *             n    sin(xn)*sqt2    cos(xn)*sqt2
 255      *          ----------------------------------
 256      *             0     s-c             c+s
 257      *             1    -s-c            -c+s
 258      *             2    -s+c            -c-s
 259      *             3     s+c             c-s
 260      */
 261                 switch(n&3) {
 262                     case 0: temp =  sin(x)-cos(x); break;
 263                     case 1: temp = -sin(x)-cos(x); break;
 264                     case 2: temp = -sin(x)+cos(x); break;
 265                     case 3: temp =  sin(x)+cos(x); break;
 266                 }
 267                 b = invsqrtpi*temp/sqrt(x);
 268         } else {
 269             u_int32_t high;
 270             a = __ieee754_y0(x);
 271             b = __ieee754_y1(x);
 272         /* quit if b is -inf */
 273             GET_HIGH_WORD(high,b);
 274             for(i=1;i<n&&high!=0xfff00000;i++){
 275                 temp = b;
 276                 b = ((double)(i+i)/x)*b - a;
 277                 GET_HIGH_WORD(high,b);
 278                 a = temp;
 279             }
 280         }
 281         if(sign>0) return b; else return -b;
 282 }